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Guidance

In the expression
, the
is called the
base
and the
is called the
exponent
.
Exponents
are often referred to as
powers
. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

.

There are many rules that have to do with exponents (often called the
Laws of Exponents
) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn a rule that has to do with raising a power to another power.

RULE: To raise a power to a new power, multiply the exponents.

Example A

Evaluate
.

Solution:
.

Example B

Simplify
.

Solution:
.

Example C

Evaluate
.

Solution:
.

Example D

Simplify
.

Solution:
.

Concept Problem Revisited

. Notice that the power rule applies even when a number has been raised to more than one power. The overall exponent is 24 which is
.

Guided Practice

You know you can rewrite
as
and then calculate in order to find that
. This concept can also be reversed. To write 32 as a power of 2,
. There are 5 twos; therefore,
. Use this idea to complete the following problems.

1. Write 81 as a power of 3.

2. Write
as a power of 3.

3. Write
as a power of 2.

Answers:

1.

There are 4 threes. Therefore

2.

There are 2 threes. Therefore
.

Apply the law of exponents for power to a power-multiply the exponents.

Therefore

3.

There are 2 twos. Therefore

Apply the law of exponents for power to a power-multiply the exponents.

Apply the law of exponents for power to a power-multiply the exponents.

Therefore

Explore More

Simplify each of the following expressions.

True or false:

True or false:

Write 64 as a power of 4.

Write
as a power of 2.

Write
as a power of 3.

Write
as a power of 3.

Write
as a power of 5.

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